## FANDOM

78 Pages

A theory $T$ with home sort $M$ is said to be dp-minimal if it satisfies one of the following equivalent conditions:

• Whenever $I$ and $I'$ are two mutually indiscernible sequences and $a$ is a singleton from the home sort, one of $I$ or $I'$ is indiscernible over $a$.
• Whenever $\langle b_i \rangle_{i \in I}$ is an indiscernible sequence and $a$ is a singleton from the home sort, there is some $i_0 \in I$ such that $\langle b_i \rangle_{i > i_0}$ and $\langle b_i \rangle_{i < i_0}$ are $a$-indiscernible.
• The home sort has dp-rank 1.

TODO: check that these definitions are correct.

If a theory is dp-minimal, it is NIP, and in fact strongly dependent. Strongly-minimal, o-minimal, C-minimal, and p-minimal theories are all dp-minimal, as are theories of VC-density 1. (TODO: check these claims.)

There is more generally a notion of dp-rank, and a set is dp-minimal if and only if it has dp-rank 1.

Expanding the theory by naming constants preserves dp-minimality.